Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I

by Edited by Alexander S. Kechris , Benedikt Löwe , John R. Steel

• ISBN13: 9780521899512
• ISBN10: 0521899516
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-09-15
• Publisher: Cambridge University Press

Summary

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research developments since the publication of the original volumes. Focusing on the subjects of 'Games and Scales' (Part 1) and 'Suslin Cardinals, Partition Properties, and Homogeneity' (Part 2), each of the two sections is preceded by an introductory survey putting the papers into present context. This volume will be an invaluable reference for anyone interested in higher set theory.

Author Biography

Alexander S. Kechris is Professor of Mathematics at the California Institute of Technology. He is the recipient of numerous honors, including the J.S. Guggenheim Memorial Foundation Fellowship and the Carol Karp Prize of the Association for Symbolic Logic, and is a member of the Scientific Research Board of the American Institute of Mathematics. Benedikt Loumlet;we is Universitair Docent in Logic and Scientific Director of the Graduate Programme in Logic at the Institute for Logic, Language and Computation of the Universiteit van Amsterdam. He is an editor of the Journal of Logic, Language and Information and managing editor for Tbilisi Mathematical Journal. He is a board member of the DVMLG and the EACSL. John R. Steel is Professor of Mathematics at the University of California, Berkeley. Prior to that, he was a professor in the mathematics department at UCLA. He is a recipient of the Carol Karp Prize of the Association for Symbolic Logic and of a Humboldt Prize. Steel is a former Fellow at the Wissenschaftskolleg zu Berlin and the Sloan Foundation.

Table of Contents

Preface	p. ix
Games and Scales
Games and scales. Introduction to Part I	p. 3
Notes on the theory of scales	p. 28
Propagation of the scale property using games	p. 75
Scales on <$$$>-sets	p. 90
Inductive scales on inductive sets	p. 94
Scales on coinductive sets	p. 102
The extent of scales in L(R)	p. 110
The largest countable this, that, and the other	p. 121
Scales in L(R)	p. 130
Scales in K(R)	p. 176
The real game quantifier propagates scales	p. 209
Long games	p. 223
The length-¿1 open game quantifier propagates scales	p. 260
Suslin Cardinals, Partition Properties, Homogeneity
Suslin cardinals, partition properties, homogeneity. Introduction to Part II	p. 273
Suslin cardinals, ¿-Suslin sets, and the scale property in the hyperprojective hierarchy	p. 314
The axiom of determinacy, strong partition properties, and nonsingular measures	p. 333
The equivalence of partition properties and determinacy	p. 355
Generic codes for uncountable ordinals, partition properties, and elementary embeddings	p. 379
A coding theorem for measures	p. 398
The tree of a Moschovakis scale is homogeneous	p. 404
Weakly homogeneous trees	p. 421
Bibliography	p. 439
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